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Abstract
We demonstrate a new method to determine the nuclear magnetic field of the spin-exchange optically pumped noble gas in a self-compensated atomic comagnetometer based on the steady-state AC response. The result shows that it has higher resolution and precision than a previous method based on the transient process. Furthermore, a convergence frequency is observed in the low-frequency region and its parameter dependence is studied simulatively, which may inspire further research into its relationship with the strong suppression mechanism of the self-compensation ability for the low-frequency magnetic field. We also prove that this method can be developed for suppression of residual main magnetic field to improve the systematic stability of the comagnetometer.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The self-compensated atomic comagnetometer [1], which contains a system of interacting alkali-metal and noble-gas spin ensembles, has been utilized for tests of fundamental symmetries [2,3], spin-dependent force [4–6], and inertial rotation as a high-precision gyroscope [7–10]. For these applications, it is essential to optimize the sensitivity of the comagnetometer by maximizing the response signal and suppressing various forms of noise. Magnetic noise is one of the important aspects; for magnetic noise, in addition to improving the performance of the magnetic shielding system outside the vapor cell [11], efforts can be invested to optimize the self-compensation ability of the atomic system inside the vapor cell [8], which is the essence of its operation principle [7].
Self-compensation ability is the ability to cancel the external random magnetic field. In the vapor cell, the alkali-metal atoms are polarized by the pump light in the z-direction, and the noble-gas atoms are hyperpolarized by alkali-metal atoms through spin-exchange optical pumping (SEOP) [12–14]. Then, the two spin ensembles are coupled through one species precessing in the effective magnetic field of the other. Due to the attraction of the alkali-metal electron wave function to the noble-gas nucleus, their effective magnetic fields are enhanced over the classical magnetic fields [1], which are described as $\boldsymbol{B}^{\boldsymbol{n}}=\lambda M^n \boldsymbol{P}^{\boldsymbol{n}}$ and $\boldsymbol{B}^{\boldsymbol{e}}=\lambda M^e {\boldsymbol {P}}^{\boldsymbol{e}}$, respectively (${\boldsymbol {P}}^{\boldsymbol{n}}$ and ${\boldsymbol {P}}^{\boldsymbol{e}}$ are the polarizations of the noble-gas nuclei and alkali-metal electrons, and $M^n$ and $M^e$ are their magnetizations corresponding to full spin polarizations). In a spherical cell, $\lambda =(8\pi \kappa _0)/3$ [15], where $\kappa _0$ is the enhancement factor. In general, a comagnetometer works in the self-compensated state as follows: $B_a=-B^c=-(B^n+B^e)$, where $B_a$ is the applied main magnetic field in the $Z$-direction (defined as the direction of the pump light) and $B^c$ is referred to as the compensation point, as shown in Fig. 1(a). In this state, the nuclear magnetization can adiabatically follow a slowly changing magnetic field, and $\boldsymbol{B}^{\boldsymbol{n}}$ can cancel to first order any changes in the transverse magnetic field ($B_x$ or $B_y$), leaving the comagnetometer only sensitive to interactions that do not scale with the magnetic moments of the atoms [7]. Figure 1(b) shows an intuitive self-compensation model with a $B_x$ perturbation: as $B_x$ slowly emerges, the nuclear polarization ${\boldsymbol {P}}^{\boldsymbol{n}}$ rotates to the $X$-axis and stabilizes in the $X$-$Z$ plane; thus, $\boldsymbol{B}^{\boldsymbol{n}}$ projects on the $X$-axis, and $B_x^n$ compensates for $B_x$. Because the total transverse magnetic field experienced by the electrons is zero, ${\boldsymbol {P}}^{\boldsymbol{e}}$ can stabilize along the $Z$-axis, unaffected by $B_x$.
Fig. 1. An intuitive model of self-compensation. (a) Alkali-metal and noble-gas atoms achieve polarization equilibriums, and the comagnetometer is set to the self-compensated state with $B_a=-B^c=-(B^n+B^e)$. (b) In this state, the nuclear polarization ${\boldsymbol {P}}^{\boldsymbol{n}}$ and the magnetization $\boldsymbol{B}^{\boldsymbol{n}}$ can adiabatically follow a slowly changing magnetic field, and $B_x^n$ will cancel the slowly emerging $B_x$, leaving the electron polarization ${\boldsymbol {P}}^{\boldsymbol{e}}$ unaffected.
In practice, the self-compensation mechanism is more complex than this intuitive model, and the effect of $B_x$ (or $B_y$) cannot be completely suppressed. The extent of this suppression is expressed as the suppression factor, which depends on $B^n$ and $B^e$ [16]. Thus, these two parameters are important for characterizing the self-compensation ability, and a precision measurement is meaningful for researches on magnetic noise suppression and sensitivity optimization.
Based on the coupling dynamics of the two spin ensembles, the magnitudes of $B^n$ and $B^e$ can be measured. When $B_a$ approaches $-B^n$, the magnetic field experienced by the electrons is close to zero and much smaller than that experienced by the nuclei. This can be ensured when $\vert B^n-B^e \vert \gg 0$, which is usually satisfied under typical experimental conditions [8,17,18]. Therefore, even though the gyromagnetic ratio $\gamma ^e$ of the electron is about three orders of magnitude larger than that of the nucleus $\gamma ^n$, the precession frequencies of the two spin ensembles can be brought into resonance and can dramatically shorten the nuclear transverse spin relaxation rate, which results in fast damping of the transient process [19]. By identifying which $B_a$ leads to the fastest decay (FD) of the transient response to a $B_y$ square wave modulation, $B^n$ can be determined, and then $B^e$ can be determined by $B^e=B^c-B^n$; $B^n$ and $B^e$ obtained by this method have been utilized to fit the suppression factor of the system response to a sinusoidal magnetic field $B_x$ to optimize the self-compensation ability of a K-Rb-$^{21}$Ne comagnetometer [8]. The FD is the only $B^n$ determination method reported so far. However, within the range of at least several nT of $B_a$ near $B_a=-B^n$, the response signals are almost identical so that it is impossible to determine which $B_a$ corresponds to the fastest decay. Thus, the measurement resolution, and consequently, the measurement precision of the FD method are not very good. The schematic of this method can be found in Ref. [8].
In this study, based on a K-Rb-$^{21}$Ne comagnetometer, a characteristic point in the steady-state AC response of the system to a $B_x$ sinusoidal modulation is found, and it is demonstrated both theoretically and experimentally that this point can be utilized for the $B^n$ (and $B^e$) measurement. Compared to the FD, the measurement point of this method is much more obvious and easier to identify, and a measurement resolution on the order of 10$^{-1}$ nT can be achieved. Accordingly, $B^n$ and $B^e$ can be determined with higher precision. Furthermore, a convergence frequency $f_c$ in the low-frequency region is found: when the $B_x$ modulation frequency is less than $f_c$, the measurement result is frequency-dependent, while when greater than $f_c$, a converged stable $B^n$ value can be obtained in a wide frequency range. This is an interesting phenomenon. In this work, the parameter dependence of $f_c$ is simulated and found to dominantly depend on $B^n$ and $B^e$; thus, this phenomenon may inspire further research into its relationship with the strong suppression mechanism of the self-compensation ability for the low-frequency magnetic field. In addition, we also prove that this method can effectively monitor the drifting $B^n$ and can be developed for $\delta B_z$ suppression to improve the systematic stability of the comagnetometer. This method of measuring $B^n$ and $B^e$ with higher resolution and precision is useful for finding the optimal self-compensated state, increasing the magnetic noise suppression ability and improving the sensitivity of the comagnetometer.
2. Theory
In a self-compensated atomic comagnetometer, to suppress the relaxation due to spin-exchange collisions between alkali-metal atoms, a high-density alkali-metal vapor cell is adopted [20], which results in an optically thick medium; in this case, the strong absorption of the pump light produces a significant polarization inhomogeneity in the cell and imposes a limit on the efficiency of optical pumping [21]. Therefore, a hybrid spin-exchange optical pumping technique that uses a mixture of K and Rb has been developed to overcome this difficulty [12,13]. In this study, with a K-Rb-$^{21}$Ne comagnetometer, the low-density K atoms, which are optically thin, are first polarized in the $Z$-direction by circularly polarized pump light; then, through spin-exchange interactions, the high-density Rb atoms are pumped by K atoms, and the $^{21}$Ne atoms are hyperpolarized by Rb atoms [2]. After a period of time, K, Rb, and $^{21}$Ne will reach their respective polarization equilibria, with stable effective magnetic fields. At typical densities, the effective magnetic field of the K atoms is much smaller than that of the Rb atoms; thus, the coupling between K-$^{21}$Ne can be ignored, and the K-Rb-$^{21}$Ne comagnetometer can be simply represented by a Rb-$^{21}$Ne comagnetometer, which is also the case in previous studies [2,22].
Complex density matrix theory is required for a full treatment of the spin evolution of the atomic ensembles [14]. Felix Bloch [23] has suggested that in many cases, it can be described with phenomenological rate equations to significantly simplify the mathematics. The Rb and $^{21}$Ne spin ensembles in a K-Rb-$^{21}$Ne comagnetometer couple together through one species precessing in the effective magnetic field of the other [24]; then, the full Bloch equations can be described as [7,25]
Equations (1) and (2) constitute a 6$\times$6 nonlinear system that is difficult to solve analytically. When a small transverse excitation is applied, we have the transverse polarization components $P_\perp ^e\ll P^e$ and $P_\perp ^n\ll P^n$. Thus, it is a fairly good approximation to assume that $\partial {P_z^{e,n}}/\partial t=0$ and the longitudinal components $P_z^e$ and $P_z^n$ are nearly constant: $P_z^e\approx P^e$ and $P_z^n\approx P^n$ (similarly, we have $B_z^e\approx B^e$ and $B_z^n\approx B^n$). The resulting 4$\times$4 system for the transverse components of the electron and nuclear polarizations becomes linear, while the main features that determine the behavior of the coupled dynamics up to high accuracy are retained [26,27]. It is convenient to write the linear system in matrix form:
whereWhen a transverse sinusoidal magnetic field $B_x=B_{x0} exp(-i\omega t)$ is applied to the system, the $X$-component of the electron polarization $P_x^e$, which is proportional to the output response signal, can be obtained by solving Eq. (3), taking the real part, we have
Using Eq. (5), the simulation of the dependences of the $B_a{\sim}A$ relationship on various parameters is shown in Fig. 2. We can see that the $A$ curve has a minimum, which is the characteristic point for the $B^n$ measurement. By analyzing the effect of each parameter, we find that when the modulation frequency $f=\omega /2\pi$ is greater than the convergence frequency $f_c$, only changes in $B^n$ and $L_z$ can cause the minimum point to shift along the horizontal axis (in the simulation of this section, $B^n$ is set to 100 nT and $L_z$ is set to 0 nT, except for the $B^n$ and $L_z$ dependences). In addition, the change in $L_z$ also affects the symmetry of the curve. For the other parameters, $R_{tot}$ mainly affects the distance between the two peaks in the horizontal direction; $B^e$, $D^e$, and $B_{x0}$ only affect the magnification of the curve along the vertical axis, while $R_{tot}^n$, $D^n$, and $Q(P^e)$ have no influence on the curve (The small influence of $Q(P^e)$ on the height of the curve is difficult to distinguish and thus is ignored; for clarity, the offsets of the $R_{tot}^n$ and $Q(P^e)$ dependences are increased and decreased by 0.3, respectively). The $f$ dependence is somewhat extraordinary, as shown at the bottom of Fig. 2: in the low-frequency region, the minimum is $f$-dependent, while it tends to converge as $f$ increases to a critical value, which is referred to as the convergence frequency $f_c$ in this text; in the high-frequency region, the minimum is fixed at $B_a=-B^n$ and immune to changes in $f$ (Thus, the other simulation results in Fig. 2 are all obtained at a relatively high frequency of $f=6$ Hz). In addition, we can see that as the frequency increases above a few hundred Hz, the curve becomes flatter and flatter, and the minimum becomes increasingly difficult to distinguish, which is also observed in the experiment. Thus, in the following experiment, the frequency range is between 0.5$\sim$500 Hz.
For a further theoretical verification, an analytic solution at the minimum is required. In principle, the solution can be obtained by solving for $B_a$ when $\partial A/\partial B_a =0$ using Eq. (5). While the equation is too complicated to obtain an explicit analytic solution, even if it exists, the solution cannot be illuminating. Therefore, we have explored other methods and found an approximation with a relative error on the order of $10^{-3}$ (the simulation result shows that this small error is mainly related to $B^n$). This approximation method is simple: when $\gamma ^n$ is assumed to decrease, the minimum of the curve will move vertically down, as shown in Fig. 3(b) with $f=6$ Hz. When $\gamma ^n$ decreases to zero, the minimum point just falls on the horizontal axis. When $f<f_c$, the trends are different. For example, in Fig. 3(a) with $f=1$ Hz, as $\gamma ^n$ decreases, the curve no longer drops vertically, while it is useful that when $\gamma ^n$ decreases to zero, the minimum also falls at $B_a=-B^n$ on the horizontal axis. Then, a simplification can be made by setting $\gamma ^n=0$ and solving for $B_a$ in $A=0$. A concise analytic solution at the minimum can be obtained as
which is consistent with the simulation result.Fig. 3. Simulation of the $\gamma ^n$ dependence of the $B_a{\sim}A$ relationship. (a) $f=1$ Hz. (b) $f=6$ Hz.
Here, we represent the $^{21}$Ne nuclear magnetic field term containing the light shift $L_z$ as $\widetilde {B^n}$, which is the parameter we directly measure in this experiment. (Since $L_z$ is equivalent to a magnetic field and affects the spin precession frequency, the measured result from the FD is also $\widetilde {B^n}$. $L_z$ and $B^c$ can be determined using a $B_y$ square wave modulation (see the next section). Thus, both parameters $B^n$ and $B^e$ can be determined by this minimum point.
3. Experimental setup and procedure
The schematic of the experimental setup is similar to our previous apparatus [29,30], as shown in Fig. 4. The comagnetometer contains a 10-mm-diameter spherical aluminosilicate glass vapor cell with a small droplet of K-Rb mixture, approximately 1000 Torr of $^{21}$Ne (70$\%$ isotope enriched), and approximately 50 Torr nitrogen gas for quenching. The cell is heated by AC-driven heating coils glued onto a boron-nitride oven, and the temperature is stabilized by a proportional integral derivative (PID) control algorithm. Two layers of 2-mm-thick high-permeability $\mu$-metal shield and a layer of 6-mm-thick ferrite shield [29] are utilized to attenuate the ambient magnetic field, and the residual magnetic field is further compensated by a set of three-axis Helmholtz coils. The temperature of the ferrite is kept well below its Curie temperature of approximately 493 K by filling thermal insulation foam between the layers. The magnetic field drift inside the shields is measured to be as small as 0.2 pT [31], thus the bias drift would not affect the results of nuclear magnetic fields measurement. The K atoms are polarized along the Z-axis by circularly polarized pump light with its wavelength tuned to the D1 resonance of potassium. The X-component of the Rb polarization $P_x^e$, which reflects the precession of the coupled spin ensembles, is measured by the optical rotation of off-resonant (0.4 nm away from the absorption center of the Rb D1 line) linearly polarized probe light. The pump and probe light intensities are stabilized by a laser intensity stabilization system, which mainly consists of a noise eater (NE) and a National Instruments PXI (NI PXI) system. The probe light is modulated by a photoelastic modulator (PEM), and the photoelectric conversion signal of the photodetector (PD) is demodulated by a lock-in amplifier to acquire the ultimate output signal [32].
Fig. 4. Experimental setup of the K-Rb-$^{21}$Ne comagnetometer (not to scale). ISO, isolator; NE, noise eater; NI PXI, National Instruments PXI system; PDA, photodiode amplifier; PD, photodetector; PBS, polarizing beam splitter; BE, beam expander; GT, Glan-Thompson polarizer; PEM, photoelastic modulator.
In this study, we use naturally abundant Rb, and the mole fraction ratio of Rb in the K-Rb mixture is approximately 0.97. The vapor temperature is set to 458 K to obtain a typically high Rb vapor density of approximately 4.8$\times 10^{14}$ cm$^{-3}$. The comagnetometer operates at the compensation point $B^c$ (referred to as operating point in this text), whose magnitude is determined by the pump light intensity $P$. In this experiment, at different pump light intensities, we measure the $^{21}$Ne nuclear magnetic field $\widetilde { B^n}$ at different operating points. After the system reaches polarization equilibrium at a pump light intensity, the three magnetic field components are first zeroed. Then, the compensation point $B^c$ and the total effective light shift $L_z$ are measured by fitting the response to a $B_y$ square wave modulation using the Lorentz curve [32]:
4. Experimental results and discussion
Under different pump light intensities, the magnitudes of $B_a$ at the minimum of the response amplitude are measured at the $B_x$ modulation frequency range of 0.5 $\sim$ 500 Hz, as shown in Fig. 5(a). The results show that the convergence frequencies $f_c$ are all less than approximately 3 Hz, leaving a wide flat frequency region for valid measurements of $\widetilde {B^n}$ at every operating point. For each experimental curve, the mean value of all measured values in the convergence region is taken as $\widetilde {B^n}$, which is listed in Fig. 5(a). The measured $\widetilde {B^n}$ values from this method are consistent with those measured by the FD method. As two examples, Fig. 5(b) and Fig. 5(c) present the details of the measurement using the $B_x$ sinusoidal modulation and the FD methods at $P=46.1$ mW/cm$^2$ and $P=23.4$ mW/cm$^2$, respectively. The light-gray shaded area represents the measurement resolution of the FD method, which is approximately 2.5 nT (this means that in the 2.5 nT range, it is impossible to identify which point decays the fastest). With this method, the amplitude variation of the response signal can be well distinguished by changing the voltage of the magnetic field $B_a$ (using the function generator) at an interval of 0.01 V. Therefore, the measurement resolution can be estimated using the coil constant in the $Z$-direction of 41.54 nT/V to be approximately 0.4 nT, which is represented by the deep-gray shaded area.
Fig. 5. (a) Measurement of the $^{21}$Ne nuclear magnetic field $\widetilde {B^n}$ at various pump light intensities using the $B_x$ sinusoidal modulation. (b) and (c) are the details of the measurement using the $B_x$ sinusoidal modulation and the fastest decay (FD) methods at $P=46.1$ mW/cm$^2$ and $P=23.4$ mW/cm$^2$, respectively.
For $P=46.1$ mW/cm$^2$, the mean value (dashed line) and the standard deviation of the 24 values of $\widetilde {B^n}$ in the convergent region are 98.52 nT and 0.06 nT, respectively; for $P=23.4$ mW/cm$^2$, they are 61.17 nT and 0.05 nT, respectively (every point represents an average of six measurements, and the error bars are given by the standard error). Thus, the measured values of $\widetilde {B^n}$ at different frequencies of the convergent region agree well with each other. For a comparison, the FD measurement is also performed at the corresponding frequencies in the convergent region. Using the mean values from the FD measurements of 98.63 nT and 61.13 nT (dash-dot lines), the relative errors of this method compared to the FD method are calculated to be 0.1$\%$ and 0.05$\%$ for $P=46.1$ mW/cm$^{2}$ and $P=23.4$ mW/cm$^2$, respectively. Therefore, the $B_x$ modulation method is valid in the whole convergent region.
The response signals for the $B_y$ square wave modulation at various pump light intensities are fitted using Eq. (7), as shown in Fig. 6, where the obtained compensation point $B^c$ and the light shift $L_z$ are listed. Thus, as two examples, $B^n$ and $B^e$ can be obtained as 98.16 nT and 52.05 nT for $P=46.1$ mW/cm$^2$ and 61.68 nT and 32.66 nT for $P=23.4$ mW/cm$^2$, respectively. As mentioned above, $L_z$ can be set to zero by adjusting the frequency of the pump light [28]; thus, the $^{21}$Ne nuclear magnetic field $B^n$ can be measured directly.
Fig. 6. Fitting result of the system response to the $B_y$ square wave modulation at various pump light intensities. The inset is the amplitude-frequency response to the $B_x$ and $B_y$ sinusoidal modulations.
At present, there are no other measurement methods reported. The resonance effect of spin precession is related to the experienced effective magnetic field. In this study, the resonance frequencies of the $^{21}$Ne nuclei and Rb electrons are on the order of 10$^{-1}$ Hz and 10$^2$ Hz, respectively. However, due to their strong coupling, the resonance peak of the $^{21}$Ne nuclei is not obvious or cannot be observed at all, as shown in the inset of Fig. 6. Even if there is a resonance peak, the contributions from nuclear and electron spins are impossible to discern [33]. In addition, the resonance frequency of the Rb electrons is also dependent on the slowing-down factor $Q(P^e)$, which is an unknown parameter. Therefore, it is impossible to measure $B^n$ and $B^e$ using the resonance effect.
To provide the convergence details of the $B_x$ modulation method in the low-frequency region, we measure the $B_a{\sim}A$ relationship in the frequency range of 0.5$\sim$3 Hz under the pump light intensity of $P=46.1$ mW/cm$^2$, as shown in Fig. 7(a) (markers). It can be seen that the measured $\widetilde {B^n}$ gradually converges with an increase in $f$. The experimental results are fitted well by Eq. (5) (solid lines), and the convergence trend is consistent with the simulation of the frequency dependence in Fig. 2.
Fig. 7. (a) Measurement of the $B_a{\sim}A$ relationship in the frequency range of 0.5$\sim$3 Hz at the pump light intensity of $P=46.1$ mW/cm$^2$; the solid lines are fitting results. (b) Simulation of the dependences of $f_c$ on $B^n$ and $B^e$.
We simulate the convergence frequency $f_c$ under various experimental conditions and find that $f_c$ dominantly depends on $B^n$ and $B^e$ (the slight association with $Q(P^e)$ can be safely ignored). It is independent of the atomic density and relaxation (and other parameters); thus, $f_c$ is not directly affected by the vapor temperature. Considering the measurement resolution of 0.4 nT, in the simulation, $f_c$ is determined when the measured $B^n(f)$ is just beginning to satisfy $|B^n (f)-B^n(\rm converged)|\leq 0.4$ nT as $f$ increases. The dependences of $f_c$ on $B^n$ and $B^e$ are shown in Fig. 7(b). At small values of $B^n$ and $B^e$, it is consistent with the corresponding experimental values of $f_c$ in Fig. 5(a). The typical values of $B^n$ and $B^e$ are less than 900 nT and 150 nT, respectively, in which $f_c$ is found to generally increase with an increase in $B^n$ or $B^e$, as shown in Fig. 7(b). The maximum of $f_c$ is only 23 Hz; thus, a wide convergence region for valid measurements can be ensured.
The convergence frequency $f_c$ is directly determined by $B^n$ and $B^e$, which are important parameters for the characterization of the self-compensation ability. In addition, simulation study shows that when the modulation frequency $f$ gradually decreases from 0.5 Hz to zero, the magnitude of $B_a$ at the minimum would increase rapidly and converge to $B_a=-B^c$. This is the operating point of the self-compensated comagnetometer. Thus, this phenomenon is speculated to be related to the strong suppression of the coupled spin ensembles to the low-frequency magnetic field. An intuitive explanation can be described as follows. When $B_a$ compensates $B^n$, and $f$ is large enough that the $^{21}$Ne spin ensemble cannot response to the modulation, the Rb spin ensemble is effectively in zero field and free to follow the modulation field adiabatically. In this case, the magnetic resonance response of the comagnetometer is similar to that of a spin-exchange relaxation-free (SERF) atomic magnetometer based on the ground state Hanle effect [34,35], and the minimum is corresponding to the linear zero crossing of its magnetic resonance dispersive response (see below). If $f$ decreases and tends to zero, the $^{21}$Ne-spins also start to follow the modulation which messes things up for the Rb-spins. In this case, the response characteristic tends to that of a typical self-compensated comagnetometer and is away from a SERF magnetometer, and the minimum where the response amplitude is most strongly suppressed falls at the compensation point $B_a=-B^c$, i.e. the operating point. If the modulation frequency is too large (e.g. $f>500$ Hz under the experimental conditions of this study), the Rb spin ensemble cannot follow adiabatically, and the response signal becomes too weak to be used for $B^n$ measurement (see the $f$ dependence in Fig. 2). According to above analysis, $f_c$ reflects the effective frequency range of the self-compensation ability to suppress external magnetic disturbances, and this ability can be improved by increasing $f_c$ (increasing $B^n$ and $B^e$ according to Fig. 7(b)). While there is no clear boundary, and the value of $f_c$ depends on the requirement of convergence accuracy. Further research on $f_c$ will deepen the understanding for the mechanism of this ability and help suppress the magnetic noise of the comagnetometer.
The nuclear magnetic field $B^n$ is constantly drifting on account of the drifting nuclear spin polarization, which leads to a nonzero residual magnetic field $\delta B_z$ and affects the systematic stability of the comagnetometer [32]. This study can also be developed for $\delta B_z$ suppression. An example is shown in Fig. 8. After the system reaches equilibrium polarization at $P=23.4$ mW/cm$^2$, we change the pump light intensity to $P=46.1$ mW/cm$^2$. Then, the Rb electron polarization reaches a new equilibrium immediately, with $B^e$ changing from 32.66 nT to 52.05 nT simultaneously. While the $^{21}$Ne nuclear polarization requires several hours to reach a new equilibrium, with $B^n$ slowly changing from 61.68 nT to 98.16 nT. Thus, we use this process to simulate the drift of $B^n$.
Fig. 8. Suppression of the residual magnetic field $\delta B_z$ using the $B_x$ sinusoidal modulation method. Inset (a) shows that the measured $B^n$ by this method gradually approaches the new equilibrium. Inset (b) shows the differences (green triangles) between $B^c$ and $B^n$, which are measured by the traditional and $B_x$ modulation methods, respectively.
Before the experiment, the residual magnetic field and the light shift $L_z$ are zeroed. The measurement order of zero in Fig. 8 corresponds to the pump light intensity being just switched, where $B^e$=52.05 nT, $B^n=61.68$ nT, and $B^c$=113.73 nT. Approximately ten minutes later, $B^n$ is measured for the first time by the $B_x$ sinusoidal modulation to be 68.10 nT, and then the compensation point can be determined to be 120.15 nT using $B^e$=52.05 nT. The compensation point $B^c$ is also measured by the traditional method (Eq. (7)) to be 120.65 nT. Similarly, this procedure is repeated until $B^n$ approaches the new equilibrium. As $B^n$ changes constantly throughout the experiment, the $B_x$ modulation measurement is performed before and after the traditional measurement each time. Then, the average of the two measurements is adopted to compare with the traditional measurement, and the difference is used to obtain the error bar. From Fig. 8, we can see that the result of this method (red circles) is consistent with that of the traditional method (black diamonds), and the validity of this method for zeroing $\delta B_z$ has been proven by many similar experiments in other vapor cells. The inset (a) in Fig. 8 shows that the measured $B^n$ by this method gradually approaches the new equilibrium. The inset (b) in Fig. 8 shows the differences (green triangles) between $B^c$ and $B^n$, which are measured by the traditional and $B_x$ modulation methods, respectively. We can see that $B^c$(Tr.)-$B^n (B_x)$ fluctuates in the range of approximately 2 nT, which is attributed to the inconsistency between the results of the $B_x$ modulation method and the traditional method during the gradual increase of $B^n$, because they are not measured at the same time. As $B^n$ gradually approaches equilibrium, the fluctuation gradually decreases. Despite the fluctuation, it is obvious that $B^c$(Tr.)-$B^n (B_x)$ is essentially constant, corresponding to a fixed value of $B^e=52.05$ nT (dashed line), which proves that this method can effectively monitor the drifting $B^n$.
Because the FD and traditional $\delta B_z$ suppression methods employ the transient response to a $B_y$ square wave modulation, it takes a relatively long time to wait for the signal to decay to the steady state before each measurement. In this study, the methods for the $B^n$ measurement and $\delta B_z$ suppression are based on the steady-state response to a $B_x$ sinusoidal modulation. Thus, the process takes less time, and it is easier to realize closed-loop control of the $\delta B_z$ suppression.
As discussed above, the magnetic resonance response of the comagnetometer at $f>f_c$ is similar to that of a SERF atomic magnetometer. And the out-of-phase term in Eq. (4) presents a good dispersive line shape, whose linear zero crossing corresponds to the minimum $B_a=-B^n-L_z$. This gives an explanation of the parameter dependences in Fig. 2: the magnetic field $B^n+L_z$ measured by the Rb magnetometer corresponds to the zero crossing, while $L_z$ and $R_{tot}$ correspond to the symmetry and the linewidth of the dispersive signal, respectively. Therefore, the $B^n$ measurement in this study can be improved by performing in a similar way as a modulated SERF atomic magnetometer based on the ground state Hanle effect [34].
To evaluate the improvement of statistical uncertainty of the new $B^n$ measurement method compared to the FD method, the comparison is scaled to the same measurement time. The time consuming of a single measurement is about 10 s and 30 s for the $B_x$ modulation method and the FD method, respectively. Given the same measurement time of 300 s, the $B_x$ method achieves a statistical uncertainty of 0.02 nT compared to the FD method that achieves 0.25 nT. In another way, in order to reach the same statistical uncertainty of 0.05 nT, the measurement time improves from 4800 s for the FD method to 60 s. Thus, the precision of the $B^n$ measurement is improved compared to the FD method. Increasing the number of measurements can reduce the standard error of the FD method, and the mean value would approach the midpoint of the uncertainty range of several nT. However, because the dependence of the decay time on the applied magnetic field $B_a$ is not simply linear, systematic error may exist (and lead to a slight discrepancy between the two methods; see Fig. 5). The systematic error of this method is due to the light shift $L_z$, which is also an important systematic error of the comagnetometer (and also of the FD method) and should be zeroed. A previous study [27] provided an effective method to suppress the total light shift $L_z$ of a K-Rb mixture based on spin-exchange collisions between the two species. Given 12 measurements in a measurement time of about 3600s (manual measurements), $L_z$ can be zeroed by this method with a statistical uncertainty of 0.12 nT (including the contribution from frequency fluctuation of the pump light). In the same time scale, the statistical uncertainties of $B_x$ modulation method and the FD method for $B^n+L_z$ measurement are 0.006 nT and 0.06 nT, respectively. Thus, the accuracies of $B^n$ measurement by both the two methods are mainly due to the accuracy of $L_z$ measurement. If the pump light frequency is well stabilized and the operation method is optimized, the new $B^n$ measurement method may achieve a higher accuracy, while the FD method is limited by itself.
5. Conclusion
We investigated a new method to determine the nuclear magnetic field $B^n$ of the spin-exchange optically pumped noble gas in a self-compensated atomic comagnetometer, which utilizes a characteristic point in the steady-state system response to a sinusoidal magnetic field $B_x$. Based on the Bloch equations, the relationship between this characteristic point and $B^n$ was both simulated and formulated. In a K-Rb-$^{21}$Ne comagnetometer, this method has been experimentally demonstrated to have higher resolution and precision than a previous method based on the fastest decay of the transient system response to a square wave magnetic field $B_y$.
In the low-frequency region of the $B_x$ modulation, a convergence frequency $f_c$ has been found: for $f<f_c$, the measured value of $B^n$ is frequency-dependent, while for $f\geq f_c$, a converged stable value of $B^n$ can be obtained, which is independent of $f$. Simulation results show that $f_c$ dominantly depends on $B^n$ and $B^e$, which are important parameters for the characterization of the self-compensation ability. Thus, this phenomenon may inspire further research into its relationship with the strong suppression mechanism of the self-compensation ability for the low-frequency magnetic field. The systematic stability of the comagnetometer is affected by the nonzero residual magnetic field $\delta B_z$ due to the drifting nuclear magnetic field $B^n$. We also proved that this method can effectively monitor the drifting $B^n$ and can be developed for $\delta B_z$ suppression, which is faster and easier to realize closed-loop control than the traditional method based on the transient process. In addition, after $B^n$ is determined, the electron magnetic field $B^e$ can be determined using the self-compensation condition, and the equilibrium polarizations of the alkali-metal and noble-gas atoms can be further obtained with known atomic densities (e.g. the Rb density can be determined using spin-exchange collision mixing of the K and Rb light shifts [36]).
This method can also be applied to other interacting atomic pairs, such as K-$^3$He and Cs-$^{129}$Xe, and the measurement of $B^n$ and $B^e$ with higher resolution and precision is useful for finding the optimal self-compensated state, increasing the magnetic noise suppression ability, and improving the sensitivity of the comagnetometer.
Funding
National Key Research and Development Program of China (2016YFB0501600, 2017YFB0503100); National Natural Science Foundation of China (61673041, 61703025, 61721091, 61773043); National Science Fund for Distinguished Young Scholars (61925301).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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